Une généralisation du triangle de Pascal et la suite A007306

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpinski gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we describe and study the first properties of the subset of [0, 1]×[0, 1] associated with this extended Pascal triangle modulo a prime p. We consider a sequence (S(n))n≥0 counting the number of positive entries on each row of the generalized Pascal triangle. By introducing a convenient tree structure, we provide a recurrence relation for (S(n))n≥0, we prove that the sequence is 2-regular, give a linear representation and make the connection with the sequence of denominators occurring in the Farey tree